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This arXiv paper reframes Transformer training as an optimal control problem, lifting the dynamics to probability measures to form a fully observed MDP and proving existence of globally optimal policies under compactness assumptions. It is theoretical work: it proposes a triply quantized training procedure and shows near-optimality, but offers no empirical benchmarks against gradient-based training. Readers should treat the claim of a "robust alternative" to gradient descent as unvalidated until experiments on realistic models appear.
An Optimal Control Approach To Transformer Training
arXiv:2603.09571v2 Announce Type: replace Abstract: In this paper, we develop a rigorous optimal control-theoretic approach to Transformer training that respects key structural constraints such as (i) realized-input-independence during execution, (ii) the ensemble control nature of the problem, and (iii) positional dependence. We model the Transformer architecture as a discrete-time controlled particle system with shared actions, exhibiting noise-free controlled McKean--Vlasov dynamics, and we show that lifting the dynamics to probability measures produces a fully-observed Markov decision process (MDP). Positional encodings are incorporated into the state space to preserve the sequence order under lifting. Using the dynamic programming principle, we establish the existence of globally optimal policies under mild assumptions of compactness. We further prove that closed-loop policies in the lifted problem is equivalent to an initial-distribution dependent open-loop policy, which are realized-input-independent and compatible with standard Transformer training. To train a Transformer, we propose a triply quantized training procedure for the lifted MDP by quantizing the state space, the space of probability measures, and the action space, and show that any optimal policy for the triply quantized model is near-optimal for the original training problem. Finally, we establish stability and empirical consistency properties of the lifted model by showing that the value function is continuous with respect to the perturbations of the initial empirical measures and convergence of policies as the data size increases. This approach provides a globally optimal and robust theoretical alternative to gradient-based training without requiring smoothness or convexity.
Abstract:In this paper, we develop a rigorous optimal control-theoretic approach to Transformer training that respects key structural constraints such as (i) realized-input-independence during execution, (ii) the ensemble control nature of the problem, and (iii) positional dependence. We model the Transformer architecture as a discrete-time controlled particle system with shared actions, exhibiting noise-free controlled McKean--Vlasov dynamics, and we show that lifting the dynamics to probability measures produces a fully-observed Markov decision process (MDP). Positional encodings are incorporated into the state space to preserve the sequence order under lifting. Using the dynamic programming principle, we establish the existence of globally optimal policies under mild assumptions of compactness. We further prove that closed-loop policies in the lifted problem is equivalent to an initial-distribution dependent open-loop policy, which are realized-input-independent and compatible with standard Transformer training. To train a Transformer, we propose a triply quantized training procedure for the lifted MDP by quantizing the state space, the space of probability measures, and the action space, and show that any optimal policy for the triply quantized model is near-optimal for the original training problem. Finally, we establish stability and empirical consistency properties of the lifted model by showing that the value function is continuous with respect to the perturbations of the initial empirical measures and convergence of policies as the data size increases. This approach provides a globally optimal and robust theoretical alternative to gradient-based training without requiring smoothness or convexity.
Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2603.09571 [cs.LG]
(or arXiv:2603.09571v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2603.09571 Focus to learn more arXiv-issued DOI via DataCite
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